Question

الريم
use division algorithom to show that
square of any positive integer
is in
the form of 3p or 3p+1​

Answer 1

Answer:

Let ′a′ be any positive integer and b=3,

Then, by division algorithm a=3q+r,

for some integer q⩾0 and 0⩽r⩽3.

So, a=3q,3q+1,3q+2.

Then, the square of positive integer,

Step-by-step explanation:

a=3q

a2=(3q)2

     =9q2

     =3(3q2)

     =3p

(Where p=3q2) 

a=3q+1

a2=(3q+1)2

      =9q2+6q+1

      =3(3q2+2q)+1

      =3p+1

(Where p=3q2+2q)

a=3q+2

a2=(3q+2)2

      =9q2+12q+4

      =3(3q2+4q+1)+1

      =3p+1

(Where p=3q2+4q+1)

Since p is some positive integer.

∴ The square of any positive integer is of the form 3p or 3p+1.